Lab 1

1/4/2022

About me

A little bit about me. As you heard on Tuesday, I am a fourth year graduate student in the Psych & Brain Sciences department. (click) My research focuses on exercise and how it influences cognition. I specifically look at how the neural mechanisms of working memory and attention are modulated as a result of an acute bout of aerobic exercise.

A fun fact about me is that I love hiking! This is a photo from a while back when I hiked Cucomonga peak down in the San Bernadino mountains. Unfortunately, I haven’t been able to hike as much during grad school but I am trying to change that with the hopefully few years I have left.

I also am very passionate about statistics, especially the Bayesian approach, machine learning, and AI. One of my hobbies is trying to use reinforcement learning methods to train video game AI to perform optimally. I say hobby because I’m not that good at it.

Purpose of labs

Facilitate learning theoretical concepts covered in lecture
Answer questions about homework
Review for midterms and the final
Implement statistical analyses from lecture in RStudio

Lab 1 Outline

Cover preliminary statistics and probability concepts
Intro to R and setup

Random Variable

A variable that takes on specific values with specific probabilities.
Measurable functions that map outcomes of a stochastic process to a measurable space.
Typically denoted by a capital letter (e.g., X, Y, Z).
Denote the outcome of a coin flip as Y
- Y = 1 if Heads, else Y = 0
- p(Y=1) = 0.5

Alright lets get started by discussing random variables. As you all know, the occurrence of events out in the world, whether it be us staying remote for the whole quarter or me TAing this class well, have a degree of uncertainty. If they didn’t, then we wouldn’t even need statistics which we use to quantify that uncertainty.

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We call these occurrences random variables, because they are variables that take on specific values with specific probabilities.

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More formally, random variables are measurable functions that map outcomes of a stochastic process to a measurable space.

(click)

Typically they are denoted by a capital letter.

(click)

For instance, we can denote the outcome of a fair coin flip as this capital Y, and map it to a measurable space of 1 and 0 if the coin lands heads or tails. The probability of either outcome occurring is 0.5.

Probability distribution

Probability mass function (PMF)
- Assigns probabilities to individual values of a discrete random variable

Probablity distribution

Probability density function (PDF)
- Similar to a PMF, but instead specifies the probability that a continuous variable takes on a range of values.

Normal Distribution Notation

X ~ N(\(\mu\),\(\sigma^2\))

Most of the variables we work with in research are continuous, in which case the probability mass function becomes a probability density function or PDF.

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For instance, we could be looking at the distribution of hours undergraduate students spend studying. It likely would take on that standard bell-shaped curve we are so used to seeing, or in other words have a normal probability density function.

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Throughout this presentation I’m going to write some things in statistical notation. This may appear frightening at first, but my goal is to expose you all to it now so that you don’t feel blind sided if you decide to take more advanced statistics course in other departments. Here, this notation is read as the random variable X has a normal probability distribution with a mean of mu and variance of sigma squared.

Expected value and Variance

The expected value of a random variable Y is denoted as E(Y).
- Probability weighted average of all possible values.

y <- c(70,80,85,90,100)
p.y <- c(0.18,0.34,0.35,0.11,0.02)

E.y <- sum(y*p.y)
E.y

## [1] 81.45

Variance
- A measure of how disperse all possible values of a random variable are from the expected value (i.e. population or sample mean)

Multivariate Distributions

PDF \(\rightarrow\) Joint Probability Density

For the random variables X and Y, the joint pdf characterizes the probability that each X and Y takes on a set of values.

Multivariate Distributions

Variance \(\rightarrow\) Covariance

Measure of how much two random variables vary together.
Formally, is the expected value of each random variable’s deviation from its respective expected value.

cov(X,Y) = E[(X - E[X])(Y - E[Y])]
Typically represented as a matrix.

Marginal Distribution

Probability distribution of an outcome for one random variable in the presence of all other outcomes for another random variable

	x₁	x₂	x₃	x₄	p_Y(y_i)
y₁	0.125	0.0625	0.03125	0.03125	0.25
y₂	0.09375	0.1875	0.09375	0.09375	0.46875
y₃	0.28125	0	0	0	0.28125
p_X(x_i)	0.5	0.25	0.125	0.125	1

Pearson correlation & Statistical independence

Pearson correlation coefficient \(\rho\) measures the linear relationship between two random variables
Two random variables are statistically independent if the realization of one does not affect the outcome of the other.

Likelihood

Throughout this course we are going to use statistical models (e.g., regression, ANOVA) to describe patterns of variability in random variables.
These models have parameters (e.g., mean of a sampling distribution)
A likelihood function is the joint probability of observed data as a function of parameters in a statistical model.

R

Programming Tips

Google is your best friend!
More than a single way to skin a cat (code).
Learn more than one language.
Have fun!